Summit

Age 16 to 18

Challenge Level Yellow star

This neat solution came from Marcos and the result was also proved by Yatir Halevi:

By the binomial expansion:

$(1+x)^m=\sum_{t=0}^m \frac{m!}{t!(m-t)!}x^t$

This can be proved by induction on

$m$ but I won't clutter this with unnecessary proofs.

Putting in

$x= -1$ we have

$0=\sum_{t=0}^m \frac{m!}{t!(m-t)!}(-1)^t$

Dividing through by

$m!$ gives us the required result:

$\sum_{t=0}^m \frac{(-1)^t}{t!(m-t)!}=0$

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